# Flat morphism

In mathematics, in particular in the theory of schemes in algebraic geometry, a **flat morphism** *f* from a scheme *X* to a scheme *Y* is a morphism such that the induced map on every stalk is a flat map of rings, i.e.,

*f*:_{P}*O*→_{Y,f(P)}*O*_{X,P}

is a flat map for all *P* in *X*.^{[1]} A map of rings A → B is called **flat**, if it is a homomorphism that makes B a flat A-module.

A morphism of schemes *f* is a **faithfully flat morphism** if *f* is a surjective flat morphism.^{[2]}

Two of the basic intuitions are that *flatness is a generic property*, and that *the failure of flatness occurs on the jumping set of the morphism*.

The first of these comes from commutative algebra: subject to some finiteness conditions on *f*, it can be shown that there is a non-empty open subscheme *Y*′ of *Y*, such that *f* restricted to *Y*′ is a flat morphism (generic flatness). Here 'restriction' is interpreted by means of fiber product, applied to *f* and the inclusion map of *Y*′ into *Y*.

For the second, the idea is that morphisms in algebraic geometry can exhibit discontinuities of a kind that are detected by flatness. For instance, the operation of blowing down in the birational geometry of an algebraic surface, can give a single fiber that is of dimension 1 when all the others have dimension 0. It turns out (retrospectively) that flatness in morphisms is directly related to controlling this sort of semicontinuity, or one-sided jumping.

Flat morphisms are used to define (more than one version of) the flat topos, and flat cohomology of sheaves from it. This is a deep-lying theory, and has not been found easy to handle. The concept of étale morphism (and so étale cohomology) depends on the flat morphism concept: an étale morphism being flat, of finite type, and unramified.

## Properties of flat morphisms

Let *f* : *X* → *Y* be a morphism of schemes. For a morphism *g* : *Y*′ → *Y*, let *X*′ = *X* ×_{Y} *Y*′ and *f*′ = *f* × *g* : *X*′ → *Y*′. *f* is flat if and only if for every *g*, the pullback *f*′^{*} is an exact functor from the category of quasi-coherent -modules to the category of quasi-coherent -modules.^{[3]}

Assume that *f* : *X* → *Y* and *g* : *Y* → *Z* are morphisms of schemes. Assume furthermore that *f* is flat at *x* in *X*. Then *g* is flat at *f*(*x*) if and only if *gf* is flat at *x*.^{[4]} In particular, if *f* is faithfully flat, then *g* is flat or faithfully flat if and only if *gf* is flat or faithfully flat, respectively.^{[5]}

### Fundamental properties

- The composite of two flat morphisms is flat.
^{[6]} - The fibered product of two flat or faithfully flat morphisms is a flat or faithfully flat morphism, respectively.
^{[7]} - Flatness and faithful flatness is preserved by base change: If
*f*is flat or faithfully flat and*g*:*Y*′ →*Y*, then the fiber product*f*×*g*:*X*×_{Y}*Y*′ →*Y'*is flat or faithfully flat, respectively.^{[8]} - The set of points where a morphism (locally of finite presentation) is flat is open.
^{[9]} - If
*f*is faithfully flat and of finite presentation, and if*gf*is finite type or finite presentation, then*g*is of finite type or finite presentation, respectively.^{[10]}

Suppose that *f*: *X* → *Y* is a flat morphism of schemes.

- If
*F*is a quasi-coherent sheaf of finite presentation on*Y*(in particular, if*F*is coherent), and if*J*is the annihilator of*F*on*Y*, then , the pullback of the inclusion map, is an injection, and the image of*f*^{*}*J*in is the annihilator of*f*^{*}*F*on*X*.^{[11]} - If
*f*is faithfully flat and if*G*is a quasi-coherent -module, then the pullback map on global sections is injective.^{[12]}

Suppose now that *h* : *S*′ → *S* is flat. Let *X* and *Y* be *S*-schemes, and let *X*′ and *Y*′ be their base change by *h*.

- If
*f*:*X*→*Y*is quasi-compact and dominant, then its base change*f*′ :*X*′ →*Y*′ is quasi-compact and dominant.^{[13]} - If
*h*is faithfully flat, then the pullback map Hom_{S}(*X*,*Y*) → Hom_{S′}(*X*′,*Y*′) is injective.^{[14]} - Assume that
*f*:*X*→*Y*is quasi-compact and quasi-separated. Let*Z*be the closed image of*X*, and let*j*:*Z*→*Y*be the canonical injection. Then the closed subscheme determined by the base change*j*′ :*Z*′ →*Y*′ is the closed image of*X*′.^{[15]}

### Topological properties

If *f* : *X* → *Y* is flat, then it possesses all of the following properties:

- For every point
*x*of*X*and every generization*y*′ of*y*=*f*(*x*), there is a generization*x*′ of*x*such that*y*′ =*f*(*x*′).^{[16]} - For every point
*x*of*X*, .^{[17]} - For every irreducible closed subset
*Y*′ of*Y*, every irreducible component of*f*^{−1}(*Y*′) dominates*Y*.^{[18]} - If
*Z*and*Z*′ are two irreducible closed subsets of*Y*with*Z*contained in*Z*′, then for every irreducible component*T*of*f*^{−1}(*Z*), there is an irreducible component*T*′ of*f*^{−1}(*Z*′) containing*T*.^{[19]} - For every irreducible component
*T*of*X*, the closure of*f*(*T*) is an irreducible component of*Y*.^{[20]} - If
*Y*is irreducible with generic point*y*, and if*f*^{−1}(*y*) is irreducible, then*X*is irreducible.^{[21]} - If
*f*is also closed, the image of every connected component of*X*is a connected component of*Y*.^{[22]} - For every pro-constructible subset
*Z*of*Y*, .^{[23]}

If *f* is flat and locally of finite presentation, then *f* is universally open.^{[24]} However, if *f* is faithfully flat and quasi-compact, it is not in general true that *f* is open, even if *X* and *Y* are noetherian.^{[25]} Furthermore, no converse to this statement holds: If *f* is the canonical map from the reduced scheme *X*_{red} to *X*, then *f* is a universal homeomorphism, but for *X* noetherian, *f* is never flat.^{[26]}

If *f* : *X* → *Y* is faithfully flat, then:

- The topology on
*Y*is the quotient topology relative to*f*.^{[27]} - If
*f*is also quasi-compact, and if*Z*is a subset of*Y*, then*Z*is a locally closed pro-constructible subset of*Y*if and only if*f*^{−1}(*Z*) is a locally closed pro-constructible subset of*X*.^{[28]}

If *f* is flat and locally of finite presentation, then for each of the following properties **P**, the set of points where *f* has **P** is open:^{[29]}

- Serre's condition S
_{k}(for any fixed*k*). - Geometrically regular.
- Geometrically normal.

If in addition *f* is proper, then the same is true for each of the following properties:^{[30]}

- Geometrically reduced.
- Geometrically reduced and having
*k*geometric connected components (for any fixed*k*). - Geometrically integral.

### Flatness and dimension

Assume that *X* and *Y* are locally noetherian, and let *f* : *X* → *Y*.

- Let
*x*be a point of*X*and*y*=*f*(*x*). If*f*is flat, then dim_{x}*X*= dim_{y}*Y*+ dim_{x}*f*^{−1}(*y*).^{[31]}Conversely, if this equality holds for all*x*,*X*is Cohen–Macaulay, and*Y*is regular, then*f*is flat.^{[32]} - If
*f*is faithfully flat, then for each closed subset*Z*of*Y*, codim_{Y}(*Z*) = codim_{X}(*f*^{−1}(*Z*)).^{[33]} - Suppose that
*f*is flat and that*F*is a quasi-coherent module over*Y*. If*F*has projective dimension at most*n*, then*f*^{*}*F*has projective dimension at most*n*.^{[34]}

### Descent properties

- Assume
*f*is flat at*x*in*X*. If*X*is reduced or normal at*x*, then*Y*is reduced or normal, respectively, at*f*(*x*).^{[35]}Conversely, if*f*is also of finite presentation and*f*^{−1}(*y*) is reduced or normal, respectively, at*x*, then*X*is reduced or normal, respectively, at*x*.^{[36]} - In particular, if
*f*is faithfully flat, then*X*reduced or normal implies that*Y*is reduced or normal, respectively. If*f*is faithfully flat and of finite presentation, then all the fibers of*f*reduced or normal implies that*X*is reduced or normal, respectively. - If
*f*is flat at*x*in*X*, and if*X*is integral or integrally closed at*x*, then*Y*is integral or integrally closed, respectively, at*f*(*x*).^{[37]} - If
*f*is faithfully flat,*X*is locally integral, and the topological space of*Y*is locally noetherian, then*Y*is locally integral.^{[38]} - If
*f*is faithfully flat and quasi-compact, and if*X*is locally noetherian, then*Y*is also locally noetherian.^{[39]} - Assume that
*f*is flat and*X*and*Y*are locally noetherian. If*X*is regular at*x*, then*Y*is regular at*f*(*x*). Conversely, if*Y*is regular at*f*(*x*) and*f*^{−1}(*f*(*x*)) is regular at*x*, then*X*is regular at*x*.^{[40]} - Assume again that
*f*is flat and*X*and*Y*are locally noetherian. If*X*is normal at*x*, then*Y*is normal at*f*(*x*). Conversely, if*Y*is normal at*f*(*x*) and*f*^{−1}(*f*(*x*)) is normal at*x*, then*X*is normal at*x*.^{[41]}

Let *g* : *Y*′ → *Y* be faithfully flat. Let *F* be a quasi-coherent sheaf on *Y*, and let *F*′ be the pullback of *F* to *Y*′. Then *F* is flat over *Y* if and only if *F*′ is flat over *Y*′.^{[42]}

Assume that *f* is faithfully flat and quasi-compact. Let *G* be a quasi-coherent sheaf on *Y*, and let *F* denote its pullback to *X*. Then *F* is finite type, finite presentation, or locally free of rank *n* if and only if *G* has the corresponding property.^{[43]}

Suppose that *f* : *X* → *Y* is an *S*-morphism of *S*-schemes. Let *g* : *S*′ → *S* be faithfully flat and quasi-compact, and let *X*′, *Y*′, and *f*′ denote the base changes by *g*. Then for each of the following properties **P**, *f* has **P** if and only if *f*′ has **P**.^{[44]}

- Open.
- Universally open.
- Closed.
- Universally closed.
- A homeomorphism.
- A universal homeomorphism.
- Quasi-compact.
- Quasi-compact and universally bicontinuous.
- Quasi-compact and a homeomorphism onto its image.
- Quasi-compact and dominant.
- Separated.
- Quasi-separated.
- Locally of finite type.
- Locally of finite presentation.
- Finite type.
- Finite presentation.
- Proper.
- An isomorphism.
- A monomorphism.
- An open immersion.
- A quasi-compact immersion.
- A closed immersion.
- Affine.
- Quasi-affine.
- Finite.
- Quasi-finite.
- Integral.

It is possible for *f*′ to be a local isomorphism without *f* being even a local immersion.^{[45]}

If *f* is quasi-compact and *L* is an invertible sheaf on *X*, then *L* is *f*-ample or *f*-very ample if and only if its pullback *L*′ is *f*′-ample or *f*′-very ample, respectively.^{[46]} However, it is not true that *f* is projective if and only if *f*′ is projective. It is not even true that if *f* is proper and *f*′ is projective, then *f* is quasi-projective, because it is possible to have an *f*′-ample sheaf on *X*′ which does not descend to *X*.^{[47]}

## See also

## Notes

- ↑ EGA IV
_{2}, 2.1.1. - ↑ EGA 0
_{I}, 6.7.8. - ↑ EGA IV
_{2}, Proposition 2.1.3. - ↑ EGA IV
_{2}, Corollaire 2.2.11(iv). - ↑ EGA IV
_{2}, Corollaire 2.2.13(iii). - ↑ EGA IV
_{2}, Corollaire 2.1.6. - ↑ EGA IV
_{2}, Corollaire 2.1.7, and EGA IV_{2}, Corollaire 2.2.13(ii). - ↑ EGA IV
_{2}, Proposition 2.1.4, and EGA IV_{2}, Corollaire 2.2.13(i). - ↑ EGA IV
_{3}, Théorème 11.3.1. - ↑ EGA IV
_{3}, Proposition 11.3.16. - ↑ EGA IV
_{2}, Proposition 2.1.11. - ↑ EGA IV
_{2}, Corollaire 2.2.8. - ↑ EGA IV
_{2}, Proposition 2.3.7(i). - ↑ EGA IV
_{2}, Corollaire 2.2.16. - ↑ EGA IV
_{2}, Proposition 2.3.2. - ↑ EGA IV
_{2}, Proposition 2.3.4(i). - ↑ EGA IV
_{2}, Proposition 2.3.4(ii). - ↑ EGA IV
_{2}, Proposition 2.3.4(iii). - ↑ EGA IV
_{2}, Corollaire 2.3.5(i). - ↑ EGA IV
_{2}, Corollaire 2.3.5(ii). - ↑ EGA IV
_{2}, Corollaire 2.3.5(iii). - ↑ EGA IV
_{2}, Proposition 2.3.6(ii). - ↑ EGA IV
_{2}, Théorème 2.3.10. - ↑ EGA IV
_{2}, Théorème 2.4.6. - ↑ EGA IV
_{2}, Remarques 2.4.8(i). - ↑ EGA IV
_{2}, Remarques 2.4.8(ii). - ↑ EGA IV
_{2}, Corollaire 2.3.12. - ↑ EGA IV
_{2}, Corollaire 2.3.14. - ↑ EGA IV
_{3}, Théorème 12.1.6. - ↑ EGA IV
_{3}, Théorème 12.2.4. - ↑ EGA IV
_{2}, Corollaire 6.1.2. - ↑ EGA IV
_{2}, Proposition 6.1.5. Note that the regularity assumption on*Y*is important here. The extension gives a counterexample with*X*regular,*Y*normal,*f*finite surjective but not flat. - ↑ EGA IV
_{2}, Corollaire 6.1.4. - ↑ EGA IV
_{2}, Corollaire 6.2.2. - ↑ EGA IV
_{2}, Proposition 2.1.13. - ↑ EGA IV
_{3}, Proposition 11.3.13. - ↑ EGA IV
_{2}, Proposition 2.1.13. - ↑ EGA IV
_{2}, Proposition 2.1.14. - ↑ EGA IV
_{2}, Proposition 2.2.14. - ↑ EGA IV
_{2}, Corollaire 6.5.2. - ↑ EGA IV
_{2}, Corollaire 6.5.4. - ↑ EGA IV
_{2}, Proposition 2.5.1. - ↑ EGA IV
_{2}, Proposition 2.5.2. - ↑ EGA IV
_{2}, Proposition 2.6.2, Corollaire 2.6.4, and Proposition 2.7.1. - ↑ EGA IV
_{2}, Remarques 2.7.3(iii). - ↑ EGA IV
_{2}, Corollaire 2.7.2. - ↑ EGA IV
_{2}, Remarques 2.7.3(ii).

## References

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